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Gaussian process-based dynamic eigenvalues as early warning signals for neuronal firing activity

Year: 2026
Type: article
Abstract: Abrupt transitions in neuronal dynamics, including changes from resting to repetitive spiking, can be viewed as critical transitions in nonlinear systems. Anticipating these transitions from voltage time series is difficult because governing equations are often unavailable. Classical early warning signals (EWSs) based on critical slowing down (CSD) capture signatures of instability, but provide neither a stability threshold nor the underlying bifurcation mechanism. Dynamic eigenvalues (DEVs) address this limitation by estimating the dominant eigenvalue of reconstructed dynamics: the approach of the DEV modulus to one gives a quantitative warning, while its complex-plane trajectory helps distinguish bifurcation types. In conventional S-map-based DEV, the local Jacobian is estimated through locally weighted linear regression, which can produce unstable DEV trajectories when the reconstructed time series is affected by strong white noise or temporally correlated colored noise. To improve the noise robustness of DEV estimation, we couple Gaussian-process (GP) differentiation of a reconstructed one-step map with the moving-window DEV framework. The resulting GP-DEV method embeds the GP posterior-mean gradient in the companion-form Jacobian and tracks its dominant eigenvalue, thereby retaining the unit-circle warning criterion and the complex-plane signatures used to distinguish bifurcation types. The conditional GP posterior is further propagated to the eigenvalue estimates, so that each warning trajectory is accompanied by a credible interval. GP-DEV is validated on benchmark systems undergoing fold, period-doubling, and Neimark-Sacker bifurcations. Under both noise types, GP-DEV yields smaller errors than S-map-based DEV and preserves more coherent complex-plane trajectories. Applied to the Morris-Lecar (ML) neuronal model, GP-DEV detects spike-onset transitions in class I/II excitability, with trajectories consistent with saddle–node on invariant circle (SNIC) and subcritical Hopf bifurcations. Ramp-evoked electrophysiological recordings further show pre-spike stability loss and bifurcation-related signatures in biological neuronal time series. These results indicate that Gaussian-process Jacobian estimation improves DEV-based early warning while retaining bifurcation-mechanism identification.
DOI: 10.1016/j.chaos.2026.119213
Language: English
Open Access status: hybrid
Estimated APC paid: $3,970
OpenAlex ID: W7214456938
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Cites: 48
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